MEN OF MATHEMATICS points in the manifold, can be neglected. For simplicity we shall state the definition when n = 4 - giving the distance between two neighbouring points in a space of four dimensions: the distance is the square root of in which the ten coefficients gn, . . . , £34 are functions of &i&s&ijCi* For a particular choice of the g's, one 'space* is defined. Thus we might have glx = 1, £22 = *> £33 = 1? £« = — 1, and all the other g's zero; or we might consider a space ia which all the g's except g44 and g34 were zero, and so on. A space considered in relativity is of this general kind hi which all the £'s except ^11,^22^33^44 are zer°j an<i these are certain simple expressions involving xltx^^x^ In the case of an n-dimensional space the distance between neighbouring points is defined in a similar manner; the general expression contains %n(n -{- 1) terms. The generalized Pytha- gorean formula for the distance between neighbouring points being given, it is a solvable problem ua the integral calculus to find the distance between any two points of the space. A space whose metric (system of measurement) is defined by a formula of the type described is called Riemannian. Curvature, as conceived by Riemann (and before him by Gauss; see chapter on the latter) is another generalization from common experience. A straight line has zero curvature; the 'measure' of the amount by which a curved line departs from straightness may be the same for every point of the curve (as it is for a circle), or it may vary from point to point of the curve, when it becomes necessary again to express the 'amount of curvature' through the use of infinitesimals. For curved sur- faces, the curvature is measured similarly by the amount of departure from a plane, which has zero curvature. This may be generalized and made a little more precise as follows. For sim- plicity we state first the situation for a two-dimensional space, 560